Exemplary embodiments described herein relate generally to passive variable negative stiffness devices that may be used, for example, to mitigate the effects of seismic events on structures, such as civil structures, or other types of vibration or other movement (e.g., equipment).
The effects of seismic events—especially large earthquakes—can be devastating to urban areas, and often include the collapse of structures, disruption of transportation infrastructure, and the interruption of business. These effects can lead to large economic losses and even the loss of life. In the United States, several major earthquakes have occurred in California within the last fifty years, such as the 1971 San Fernando earthquake, the 1989 Loma Prieta earthquake, and the 1994 Northridge earthquake. Each of these events resulted in the collapse of large buildings or major highways, with deaths numbering less than a hundred per event (National Research Council, 2011). However, the magnitude (<7) of each of these earthquakes, and the size of the area that was impacted, was comparatively small.
In contrast, a report published in 2008 by the United States Geological Survey estimated that a magnitude 7.8 earthquake occurring on the southernmost 200 miles of the San Andreas Fault would result in 1,800 deaths and $213 billion in economic losses (building damages, non-structural damages, damage to lifelines and infrastructure, and fire losses). Furthermore, this hypothetical event does not even represent the largest earthquake that can be produced by the San Andreas Fault at the selected location.
The inelastic response of building structures combined with supplemental viscous damping has motivated research into apparent weakening for seismic response control. The term apparent weakening refers to the softening of the structure apparent stiffness through the addition of negative stiffness at a displacement that is smaller than the structure yield displacement. The resulting structure with combined positive and negative stiffness emulates yielding, thereby limiting the base shear forces and absolute accelerations. Viscous damping is also added to the structure to limit the increase in displacements that occur due to softening. The structure benefits from the yielding behavior, without incurring the damage associated with actual yielding, for structure displacements smaller than the yield displacement. While the benefits of apparent weakening may be realized through the addition of negative stiffness, it requires a passive adaptive negative stiffness device (NSD), which is not easily achieved.
In order to emulate yielding at a structure displacement that is smaller than the actual yield displacement (uy), negative stiffness should be added to the structure when it reaches a predetermined apparent yield displacement (uy′). This is represented by the dark grey line extending down from the x-axis on the plots shown in FIG. 1. If the negative stiffness added to the structure is equal to the structure positive stiffness (light grey line), then the combined structure-NSD stiffness will be zero (black line). Once the structure displacement reaches uy, yielding will occur and there will be a reduction in the structure stiffness. If the negative stiffness added to the structure remains unchanged beyond uy, the combined structure-NSD stiffness will become negative, and the structure will become unstable. Therefore, after uy has been reached, the negative stiffness should be removed.
Negative stiffness can be removed by transitioning to positive or zero stiffness. Transitioning to positive stiffness will result in a stiffening of the combined structure-NSD system as the positive stiffness from the NSD is added to the inelastic stiffness of the structure, leading to an increase in the combined stiffness beyond that of the inelastic stiffness of the structure (see FIG. 1 (a)). Transitioning to zero stiffness also leads to an increase in the stiffness of the structure-NSD system as the negative stiffness is removed and the inelastic stiffness of the structure remains (see FIG. 1(b)). However, the resulting combined structure-NSD stiffness will be less than that which occurs with the transition to positive stiffness. The transition to zero stiffness results in structure-NSD behavior that more closely emulates a yielding structure, and has been described as an ideal case that is difficult to achieve with a passive device.
Past research in apparent weakening has been based on a negative stiffness device that transitions to positive stiffness, rather than zero stiffness, after structure yielding. In particular, an adaptive negative stiffness system (ANSS) has been studied extensively through numerical and experimental investigation. The force-displacement profile of the ANSS is best represented by that shown in FIG. 1(a). Analytical modeling of the NSD component of the ANSS has been performed, and the effect of large rotations and inertia on the NSD force has been examined. An analytical model has been used to simulate the behavior of an equivalent SDOF building structure with NSD, and with and without dampers, subject to both periodic and random ground motions. It was found that for structures that remain in the elastic range, the base shear of the structure is reduced substantially and passive damping can be used to limit displacements. For yielding structures, it was found that the appropriate combination of NSD and a passive damper can significantly reduce displacements, accelerations, and base shear compared to the base structure. In laboratory studies on a fixed-base building without yielding, it was found that adding the NSD with no viscous dampers reduced the base shear by more than 30%, and the peak acceleration by more than 20%, compared to the structure with no NSD for the strong earthquakes considered. For the same building with mild yielding, the base shear and accelerations of the structure-NSD system were reduced by more than 30%. However, for severe ground motions, the system is subject to large displacements during which stiffening occurs. The stiffening results in an increase in the base shear and acceleration. The addition of the viscous damper to the structure-NSD results in consistent reductions in displacements, acceleration, and base shear by more than 20%.
In addition to the fixed-base building research, the performance of the ANSS has also been investigated for a seismically-isolated building. It was found that adding the NSD to the isolation system resulted in a reduction in the base shear force (force transmitted to the foundation), inter-story drift of the superstructure, and floor accelerations, and had little to no effect on the isolator base displacements. However, it was shown that the addition of viscous dampers to the isolation system with NSD resulted in a substantial reduction in base displacements. Additional shake table studies were performed to study the effectiveness of the ANSS for seismic isolation of a highway bridge model. The results showed that the inclusion of the NSDs can significantly reduce the shear forces in the substructure, thereby protecting the bridge piers and abutments from strong earthquakes. One of the favorable outcomes of the research was the effectiveness of the NSDs even when a flexible layer (i.e., bridge piers) is inserted between them and the foundation of the structure. Also of note was that the system achieved a non-resonant condition due to the constantly changing stiffness once the NSD is engaged. Numerical simulations on the same bridge model showed that the ANSS will not be effective for all ground motions, but that flexibility within the design parameters of the NSD may allow it to be modified to be effective for site-specific ground motions.
Another adaptive passive NSD that has been proposed for seismic protection is the rotation-based mechanical adaptive passive stiffness (RBMAP-S) device. It consists of a primary gear and two secondary gears mounted in series, but not initially in contact. The secondary gears are pre-torqued and held in place using a pawl. Rotation of the primary gear is initiated by displacement of the structure through a connecting arm. The system was designed so that the primary gear engages one of the two secondary gears after a predetermined structure displacement. Once engaged, the secondary gear disengages from the pawl and transfers the stored torque to the primary gear, which in turn transfers a force to the structure that assists its motion. Small-scale experiments and analytical equations both showed a sudden increase in the force exerted on the structure once the secondary gear engaged due to the sudden increase in torque on the primary gear. A modified version of the device including an additional precompressed spring, torsional spring, and slotted connections was proposed to obtain the desired force-displacement profile, which is best represented by FIG. 1(a).
A third adaptive passive NSD that has been recently proposed for seismic protection is the bio-inspired passive negative spring actuator (BIPNSA), which utilizes a preloaded spring attached between the first story of a structure and a roller bearing supported by a ramp on the ground floor. Lateral displacement of the upper story causes the roller bearing to move down the ramp and exert a horizontal force that assists the structure motion. Experimental validation of the concept was performed, and the same configuration was shown to be effective in small-scale shake table experiments at reducing the first floor displacement and third floor acceleration of a seismically-excited building model.
In sum, a review of the state-of-the-art in adaptive passive negative stiffness devices revealed only three that may be capable of producing the force-displacement profiles required for apparent weakening. Of these, the ANSS system was studied extensively through numerical methods and large-scale laboratory experiments. From those studies, it was revealed that the flexibility and inertia of the ANSS assembly influenced the negative stiffness of the system, and should therefore be minimized. It was also reported that improperly designed gap spring assemblies (GSAs) can lead to an undesirable force-displacement behavior of the assembly. Other issues with respect to the efficiency and reliability of the ANSS may also be related to the GSAs. The GSA forces are added to the NSD at all structure displacements, even after uy′ has been reached. As a result, the force from the precompressed vertical spring must be excessively large so that the net force produced by the NSD matches the desired force-displacement profile. Another potential issue with the GSAs is damage during an earthquake, which could occur through impact as the GSAs are continuously engaged and disengaged during operation. Damage to the GSAs would result in an unwanted increase in the negative stiffness provided to the structure, resulting in a potentially unstable condition.
Exemplary embodiments of the present invention may address some or all of the shortcomings of the known art. In particular, exemplary embodiments of the present invention may facilitate a transition to zero stiffness by using elastic devices (e.g., springs) combined with translating and rotating components. Some exemplary embodiments may allow, for example, spring sizes to be minimized. An example of the efficient design may increase the ability to realize the system in a variety of environments. Some exemplary embodiments may also allow for forces to be applied to, for example, a subject structure substantially continuously, which significantly improves the ability to account for movement in the subject structure while minimizing impact forces on the structure. For instance, some exemplary embodiments may be particularly useful in association with building and bridge structures, such as to address movement caused by seismic activity, wind, etc. or any other type of movement. Exemplary embodiments may also be used in association with other types of systems, structures, devices, etc., such as equipment that has a tendency to vibrate or experience other undesired movement during operation or for other types of movement isolation applications.
In addition to the novel features and advantages mentioned above, other benefits will be readily apparent from the following descriptions of the drawings and exemplary embodiments.